The D750 is a great camera. Keep in mind it is FX and most of your lenses are DX. They will fit and take a photo but you'll see black edges in the viewfinder and your pictures will be 10.7MP which is actually less than your D90.
About this... I’ve never tried to use a DX lens with a full frame (Nikon) body. Is this (the fact that it still works using a small portion of the sensor) just a Nikon thing? I remember hearing that Canon EF-S (DX equivalent) lenses cannot even be used at all with full frame Canon bodies.
Canon EF-S lenses are not compatible with Canon full frame DSLRs.
One of the limitations of a DSLR lens is that if the rear element can't be too close to the sensor or it will interfere with the operation of the mirror.
Crop body DSLRs have smaller mirrors. Some EF-S lenses have rear elements that protrude a little into the camera body. This is fine with a crop body mirror, but not compatible with a full frame mirror.
It turns out that allowing the rear element to be closer to the sensor can have optical advantages for certain focal lengths, allowing for a higher quality lens at a lower cost.
Therefore Canon's crop body EF-S lenses have the advantage of an optimized design, but the disadvantage that they can't be used on a full frame DSLR.